Question 1:

Consider the standard market model:

R_it = α_i + β_iR_mt + u_i∀_i = N and ∀t = 1...T

where E[u_isu_jt] = σ_ij, if s = t and 0 otherwise.

Would there be any gain in the precision of our estimators for α_i, and β_i∀i if we would use a seemingly unrelated regression model as opposed to N regressions involving ordinary least squares? Prove why or why not.

Question 2:

For the seemingly unrelated regression model in the case where the variance covariance matrix of the disturbances is unknown, we generally impose the restriction that T ≥ N, where T is the number of observations per equation and N is the number of equation in the system. Why is this restriction imposed?

Question 3:

If the disturbances across equations are not contemporancously correlated in the seemingly unrelated regression model, prove that using ordinary least squares equation by equation is just as efficient as estimating the system of equation jointly.

Question 4:

Investigate whether or not the mean of stock returns differs by the month of the year. For example, is the average return during the month of January higher or lower than the average return on other months? You are expected to perform a rigorous statistical test of a formal statistical hypothesis (Note by using SAS or EXCEL date functions you can casily compute the month of the year for each observation).

Question 5:

Given the following:

E_t-1[M_t(1 + R_it)] = 1∀i,t.

where M_t, is the aggregate marginal rate of substitution between t-1 and t, R_it, is the return on asset I from t-1 to t, and the expectation is with respect to all available information as of t -1.

Prove the following:

E_t-1 [R_it - R_zt] = β_itE_t-1[R_ct - R_zt]

where R_ct, is the return on a portfolio which is perfectly correlated with M_t, R_zt is the return on a portfolio uncorrelated with R_ct and β_it = cov_t-1(R_it, R_ct)/VAR_t-1, (R_ct)

Question 6:

Using a set of pooled time series regressions, test the mean-variance efficiency of the CRSP index relative to the size portfolios provided in the data set. You should rely on the Gibbons Ross Shaken article in performing this test. Discuss how this test can be interpreted as a test of the Sharpe-Lintner CAPM.

Question 7:
The above test makes use of a distributional assumption on returns. Derive a test of the efficiency of CRSP index which does not rely on a distributional assumption on returns. That is, test the mean variance efficiency of the CRSP index directly.

Question 8:
Another way to test the CAPM has been suggested in the literature. This alternative technique involves regressing sample mean returns on estimated betas. Run these cross- sectional regressions on the data base provided and comment on the results. You should find it instructive to do a scatter plot of sample means against estimated betas. Discuss the motivation behind such tests; 7e explain why this could be interpreted as a test the CAPM. Discuss all the various econometric problems with such tests.

Question 9:
Test the Black version of the CAPM assuming the CRSP index is the market portfolio. You should rely on Gibbons (1982) in performing this test.

Question 10:
Replicate the non parametric test suggested in Brown and Gibbons (1985) using the CRSP index and the risk less rate. Report the estimate of relative risk aversion as well as its standard error.

Now extend the model in Brown and Gibbons to the multi-asset case in the manner suggested in their appendix. Test this model using the size portfolios and the riskless rate. Report the estimate of the relative risk aversion as well as its standard error. What does the over-identifying restriction test reveal as to the validity of the model?